Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Show Recent Comments

10 Most Recent Blog Comments:

Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

March 28, 2021 at 12:03am

#1007169

Facts... Maybe?

When you want to learn reliable facts about stuff, you can always turn to... Cracked?

55 Ways You're Picturing History Wrong

Okay, maybe, maybe not. Some of these track with what I know. And no, I'm not going through all 55; you can read the article for that; they're all short blurbs because after all, it's Cracked, not The New Yorker. Which also means it's actually funny.

Ever since 1978, when it became illegal to teach history in schools, all our knowledge on the subject has come from movies, TV shows, and vague chitchat. As a result, we're all buried under misinformation. For example, someone's been spreading weird rumors about how the education system changed in 1978.

Okay, reliable except when they turn to obvious sarcasm.

The article also provides handy links for fact-checking, but of course to be thorough, one should independently verify each claim. If one can be arsed. I can't.

4. Cowboys Hats

American cowboys wore all kinds of hats. Bowler hats were most common. You'd also see sombreros or top hats.

The one time I was at a dude ranch, the only hat I had available was a Panama Jack. None of the Stetsons they sold us city folks fit my head (all those brains take up so much room, you know), so I probably have the distinction of owning the only Panama Jack with a stampede strap.

10. Cavalry Charges

In a movie, when the backup army arrives on horses, they tear through the men or orcs on foot. But in reality, horses smartly refuse to plow through soldiers bearing pikes, so cavalry divisions would dismount before fighting.

Thus once again making one wonder who the truly smart species is.

15. American Cowboys

Cowboys are a Mexican invention. There were vaqueros in Mexico decades before settlers came even to Plymouth. By the 19th century, a third of cowboys in America were Mexican -- and one quarter were Black.

Speaking of cowboys.

Also, I always wondered about that word. I mean, yeah, I knew just enough Spanish to understand what "vaquero" means (along with hola, gracias, and marijuana), but it's gotta confuse non-English-speakers. They ride horses, not cows (yes, I understand they herd cows) and they're generally too old to be "boys." Besides, "cow" is always female. Also, why is there a collective word for every kind of animal except cattle? Like, you can't point at a single cow, bull, or steer and say, "That's a cattle," because cattle is plural. You have to call it a cow, bull, or steer. You don't have that problem with mares, stallions, and geldings; you can point and go "Horsey!"

English is weird.

36. Ninjas

Ninjas didn't dress in black. They disguised themselves in whatever costumes best blended in. Even skulking at night, they didn't wear black, which would stand out like a silhouette.

Fact: Almost every photograph contains at least one ninja.

39. The Library of Alexandria

The burning of the Library of Alexandria really wasn't much of a tragedy. The library had already fallen into disrepair for centuries -- not due to fire but due to budget cuts.

Gosh, that sounds familiar.

46. Martin Luther

Luther didn't actually nail 95 theses to a church door, as cool as that would have been.

Look, this is one of those cases where it's much more amusing to accept the myth, like with George Washington and the cherry tree.

51. Pocahontas

Pocahontas was at most 12 years old when the Jamestown settlers came by. She was also bald and naked, as was customary for her tribe. Good then that she and John Smith didn't really fall in love.

No one tell Disney.

54. The Alamo

The Alamo wasn't a heroic last stand that saved Texas. The rebels only stuck around there because they ignored smart orders, and defending the place didn't help the war effort at all.

Oh, Cracked wants to get banned in Texas, I see.

Actually, I'm only including this one because my favorite drafthouse theater chain is called Alamo because they started in Texas. As I went there quite a bit before everyone started passing dangerous germs around, I became a member of their frequent-flyer club. I'm still amused at the name of the frequent-flyer club: Alamo Victory.

My local Alamo seems to have survived the pandemic and bankruptcy closings that hit some of that chain. I hope it lasts. Soon as I can, I'm going back.

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